How do we connect them so that after the c wash and rinse (copy and paste) runs it automatically runs the "TS Game 3_50 COUNTERS-1.xlsm" and then back to the copy and paste?
Without going into Error Checking, Make sure both books are Open. Note that Any Macro can check for open books and open them if needed, But why don't we wait until the code is working with that for now.

Real basic, 'cuz I have to leave now.

Put this in a Standard Module in TS Game 3_50 COUNTERS-1.xlsm 
Option Explicit
 
Sub ertert()
    Dim x, i&, j&
    With Sheets("Counter Totals")
        x = .Range("A2:CM" & .Cells(Rows.Count, 1).End(xlUp).Row).Value
    End With
    For i = 1 To UBound(x) 'The LBound of x is 0 <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
        If (x(i, 1)) = "Game" Then j = j + 1
        If (IsNumeric(x(i, 1))) * (Len(x(i, 1))) Then
            With Sheets("Game" & x(i, 1)).Columns(1).SpecialCells(2)
                .Areas(j)(.Areas(j).Count + 1, 1).Resize(, 91).Value = Application.Index(x, i, 0)
            End With
        End If
    Next i
    Workbooks("TS 4+ BB Game 3 (Macro Numbers Input)-1.xlsm").NumbersGrabber 

End Sub
And this in a Standard Module in "TS 4+ BB Game 3 (Macro Numbers Input)-1.xlsm" 
Option Explicit

Sub NumbersGrabber()
    Dim InputSht As Worksheet
    Dim DrnNumSht As Worksheet
     
    Static NextRowToUse As Long 'maintains State between calls as long as Workbook is open. With caveats.
     
    Set InputSht = Workbooks("TS Game 3_50 COUNTERS-1.xlsm").Sheets("Input")
    Set DrnNumSht = Sheets("Drawn Numbers")
     
    If NextRowToUse = 0 Then NextRowToUse = DrnNumSht.Cells(Rows.Count, "I").End(xlUp).Row
    If NextRowToUse = 1 Then Exit Sub
     

    DrnNumSht.Cells(NextRowToUse, "I").Resize(, 7).Copy 'Resize(0 Rows, 7 Columns)
    InputSht.Range("I49").Resize(, 7).Insert shift:=xlShiftDown
    NextRowToUse = NextRowToUse - 1

    'Run the etert Procedure
    'Workbooks("TS Game 3_50 COUNTERS-1.xlsm".ertert '<-------- UnComment after testing
     
End Sub
Make sure there is only one sub with the same name in any Workbook.

Notre the Workbook is specified in each Call to the other routine.